• Bulletin of the Korean Mathematical Society
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• v.38 no.3
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• pp.443-447
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• 2001

The main goal of this paper is to show the following: Let d and g be (continuous or discontinuous) linear derivations on a Banach algebra A over a complex field C such that $\alphad^3+dg$ is a linear Jordan derivation for some $\alpha\inC$. Then the product dg maps A into the Jacobson radical of A.

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.267-276
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• 2007

Injective JW-algebras are defined and are characterized by the existence of projections of norm 1 onto them. The relationship between the injectivity of a JW-algebra and the injectivity of its universal enveloping von Neumann algebra is established. The Jordan analgue of Theorem 3 of [3] is proved, that is, a JC-algebra A is nuclear if and only if its second dual $A^{**}$ is injective.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.277-283
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• 2022

Let 𝒜 be a unital Banach algebra, 𝓜 a unital 𝒜-bimodule, and 𝛿 a linear mapping from 𝒜 into 𝓜. We prove that if 𝛿 satisfies 𝛿(A)A^-1+A^-1𝛿(A)+A𝛿(A^-1)+𝛿(A^-1)A = 0 for every invertible element A in 𝒜, then 𝛿 is a Jordan derivation. Moreover, we show that 𝛿 is a Jordan derivable mapping at the unit element if and only if 𝛿 is a Jordan derivation. As an application, we answer the question posed in [4, Problem 2.6].

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.117-132
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• 2011

By developing a linear algebra program involving many different structures associated to a three-graded H*-algebra, it is shown that if L is a Lie triple automorphism of an infinite-dimensional topologically simple associative H*-algebra A, then L is either an automorphism, an anti-automorphism, the negative of an automorphism or the negative of an anti-automorphism. If A is finite-dimensional, then there exists an automorphism, an anti-automorphism, the negative of an automorphism or the negative of an anti-automorphism F : A $\rightarrow$ A such that $\delta$:= F - L is a linear map from A onto its center sending commutators to zero. We also describe L in the case of having A zero annihilator.

• Korean Journal of Mathematics
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• v.26 no.2
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• pp.285-291
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• 2018

Let $\mathcal{A}$ be a unital $C^*$-algebra. It is shown that additive map ${\delta}:{\mathcal{A}}{\rightarrow}{\mathcal{A}}$ which satisfies $${\delta}({\mid}x{\mid}x)={\delta}({\mid}x{\mid})x+{\mid}x{\mid}{\delta}(x),\;{\forall}x{{\in}}{\mathcal{A}}_N$$ is a Jordan derivation on $\mathcal{A}$. Here, $\mathcal{A}_N$ is the set of all normal elements in $\mathcal{A}$. Furthermore, if $\mathcal{A}$ is a semiprime $C^*$-algebra then ${\delta}$ is a derivation.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.331-343
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• 2024

Let ℜ be a *-algebra with unity I and a nontrivial projection P₁. In this paper, we show that under certain restrictions if a map ψ : ℜ → ℜ satisfies $$\Psi(S_1{\diamond}S_2{\cdot}{\cdot}{\cdot}{\diamond}S_{n-1}{\bullet}S_n)=\sum_{k=1}^nS_1{\diamond}S_2{\diamond}{\cdot}{\cdot}{\cdot}{\diamond}S_{k-1}{\diamond}{\Psi}(S_k){\diamond}S_{k+1}{\diamond}{\cdot}{\cdot}{\cdot}{\diamond}S_{n-1}{\bullet}S_n$$ for all S_n-2, S_n-1, S_n ∈ ℜ and S_i = I for all i ∈ {1, 2, . . . , n - 3}, where n ≥ 3, then ψ is an additive *-derivation.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.149-158
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• 2012

In this paper, we prove the superstability and the generalized Hyers-Ulam stability of Jordan *-homomorphisms between unital $C^*$-algebras associated with the following functional equation$$f(\frac{-x+y}{3})+f(\frac{x-3z}{c})+f(\frac{3x-y+3z}{3})=f(x)$$. Morever, we investigate Jordan *-homomorphisms between unital $C^*$-algebras associated with the following functional inequality $${\parallel}f(\frac{-x+y}{3})+f(\frac{x-3z}{3})+f(\frac{3x-y+3z}{3}){\parallel}\leq{\parallel}f(x)\parallel$$.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.607-618
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• 2002

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A \longrightarrowA such that D($\chi$)[D($\chi$),$\chi$]D($\chi$) $\in$ rad(A) for all $\chi$ A. In this case, we have D(A) ⊆ rad(A).

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.211-224
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• 2002

In this paper we show that if D is a continuous linear Jordan derivation on a Banach algebra A satisfying [[D($x^{n}$), $x^{n}$, $x^{n}$] $\in$ rad(A) for a positive integer n and for all x${\in}$A, then D maps A into rad(A).

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.727-739
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• 1998

We discuss dual algebras generated by a contraction and properties $({\mathbb}A_{m,n})$ which arise in the study of the problem of solving systems of the predual of a dual algebra. In particular, we study membership for the class ${\mathbb}A_1,{{\aleph}_0 }$. As some examples we consider dual algebras generated by a Jordan block.